# MATH 1130: Show Using an Honest Hilbert Proof that ⊢𝐴∧(𝐴∨𝐵)≡𝐴: Discrete Mathematics 1 Assignment, DC, Canada

University | Douglas College (DC) |

Subject | MATH 1130: Discrete Mathematics 1 |

## Question 1:

Show using an honest Hilbert proof that ⊢𝐴∧(𝐴∨𝐵)≡𝐴.

For this question, your answer must be a Hilbert proof according to Definition 1.4.5, no shortcuts (including meta theorems or absolute theorems from class/book) are allowed.

## Question 2:

(a) Prove that for any wffs 𝐴,𝐵,𝐶, D and any propositional variable 𝑝,

𝐴→(𝐵≡𝐶)⊢𝐴→(𝐷[𝑝:=𝐵]≡𝐷[𝑝:=𝐶])

(Hint: Using the deduction theorem or Post’s theorem makes this easier, although there are other proofs).

(b) Is it true that 𝐴→(𝐵→𝐶)⊢𝐴→(𝐷[𝑝:=𝐵]→𝐷[𝑝:=𝐶])? If yes, give a proof of this, and if no, find a counterexample.

## Question 3:

Consider the string ((((𝑝∨𝑟)≡𝑞)→((𝑟∧(¬𝑝))≡𝑟))→𝑟)

(a) Write a formula-calculation to show this string is a wff.

(b) Is this formula a tautology? Explain why or why not.

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info@studentsassignmenthelp.com## Question 4:

(a) Give an example of a well-formed formula 𝐴 so that we do NOT have ⊢¬𝐴. Explain why your answer works (using soundness might help).

(b) What is wrong with the following equational proof of ⊢¬𝐴?

¬𝐴

⇔⇔ (¬-introduction axiom)

𝐴≡⊥

⇔⇔ (Leibniz from prev. line, “C-part” 𝑝≡⊥)

⊥≡⊥

⇔⇔ (⊤ vs. ⊥)

⊤

## Question 5:

Use the method we discussed from the “Weak Post’s Theorem” notes to show:

⊢((𝑝∨¬𝑞∨𝑟)∧(𝑝→𝑟))→(𝑞→𝑟)

## Question 6:

Write an equational-style proof for the following: 𝑟→(𝑝∨𝑞)⊢(𝑟∨(𝑝∨𝑞))→(𝑝∨𝑞).

## Question 7:

Give proof to show

⊢((𝑝→𝑞)∧(𝑝→𝑟))→(𝑝→(𝑞∧𝑟))

(Use equational or Hilbert proof styles. Post’s theorem is not allowed).

## Question 8:

What is wrong with the following equational proof of ⊢(𝐴→𝐶)≡𝐴?

𝐴→𝐶

⇔⇔ (implication axiom)

𝐴∨𝐶≡𝐶

⇔⇔ (Leibniz+redundant true; “C-part” 𝐴∨𝑝)

𝐴∨⊤

⇔⇔ (∨-identity)

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